A differential equation (D.E.) is an
equation involving derivatives in its statement, such as $\frac{dy}{dx} = x^2$ or $\frac{dy}{dx} = \frac {x^2}y$. The study of
differential equations and their solutions is a study that often begins
in the first semester of a calculus course and can continue through much
of the study of mathematics and its application in almost every
science.

For more on differential equations see The Sensible Calculus, **IV. A. Differential Equations. Basic Concepts
and Definitions.**

In some situations a differential equation may be expressed using differentials!

For example, the equation $\frac{dy}{dx} = x^2$
might be expressed as $dy =x^2 dx$ and $\frac{dy}{dx} = \frac {x^2}y$
might be expressed as $dy =\frac{ x^2}y dx$
or as $y dy = x^2 dx$

A solution to a differential equation with one dependent variable is a
differentiable (hence, continuous) function which satisfies the
differential equation on an interval domain. The general solution to a
differential equation is the family of all functions that are solutions
to a given differential equation.

Often a differential equation has one or
more additional conditions which a solution must satisfy as well, such
as y(0) = 2 or y(3) = 4. Additional conditions often arise in
applications from information given about the beginning of a situation
("initial conditions") or conditions that must be satisfied at the
beginning and end of the process ("boundary conditions"). Such
conditions restrict the selection of solutions to only a few particular
members from the family of all solutions. A particular (or special)
solution to a differential equation is a solution that satisfies any
such additional conditions.

The interpretation of the derivative as a rate makes the study of
differential equations a significant part of the study of rates in many
contexts.

The visualization of rates and differential equations often uses a
cartesian slope field. However mapping diagrams are not used in most
calculus textbooks to visualize differential equations.

Before continuing with this section it would be helpful to review the materials in CCD.NA on
the differential.

Change the example derivative, the value of a, or initial condition value for f(a) by making an entry in the appropriate box. The value of a can also be changed by using the slider. On the cartesian graph, moving the point labeled "Move me" will change both the value of a and the initial condition value for f(a).

Checking the boxes has the following effects:

- Show Differential: Shows a second point, $x'=x+\Delta x$, on the mapping diagram determined by $\Delta x = dx$ on the slider. An arrow from $x'$ indicated the estimate of $f(x') \approx y + dy$ determined by the derivative, $\frac{dy}{dx} $. On the cartesian graph a line segment centered at the point labeled "Move me" of length $\Delta x$ visualizes a tangent line with slope $\frac{dy}{dx}$ at the point $(a, f(a)$. Moving the slider for $\Delta a$ will adjust the related fetures of the figure.
- Show Solution: Shows a solution for the differential equation as determined by GeoGebra's computer algebra and the related cartesian graph as well as the value of the solution for $x'=x + \Delta x$ on the graph and the mapping diagram.
- Show/Hide Slopefield: Shows the slope
(tangent, direction) field for the differential equation. The slider N
can be used to change the number of segments shown in the field and their lengths.

**CIS.EM.TMD**.
**Euler's Method with Tables and Mapping Diagrams.**